Project documentation
Existence theorem packaging the lexicographic construction. It produces a LinearOrder (Finset α) with two convenient properties: 1. Subset-monotonicity: A ⊆ B implies A ≤ B. 2. Singleton compatibility: comparing singleton finsets recovers the original order on α. API note: returning the order via ∃ lo allows users to avoid a global instance and...
Exact Lean statement
theorem Lex'Order_prop (α : Type*) [lo : LinearOrder α] : ∃ lo : LinearOrder (Finset α),
(∀ A B : Finset α, A ⊆ B → lo.le A B) ∧
(∀ a b : α, a ≤ b ↔ lo.le {a} {b})Formal artifact
Lean source
theorem Lex'Order_prop (α : Type*) [lo : LinearOrder α] : ∃ lo : LinearOrder (Finset α),(∀ A B : Finset α, A ⊆ B → lo.le A B) ∧(∀ a b : α, a ≤ b ↔ lo.le {a} {b}) := by use Lex'LinearOrder constructor · intro A B h unfold LE.le Preorder.toLE PartialOrder.toPreorder LinearOrder.toPartialOrder Lex'LinearOrder LexLE if heq : A = B then right constructor · exact congrArg Finset.card heq · rw [heq] else left apply Finset.card_lt_card exact Finset.ssubset_iff_subset_ne.2 ⟨h,heq⟩ · intro a b unfold LE.le Preorder.toLE PartialOrder.toPreorder LinearOrder.toPartialOrder Lex'LinearOrder LexLE simp only [SemilatticeInf.toPartialOrder, Lattice.toSemilatticeInf, SemilatticeSup.toPartialOrder, Lattice.toSemilatticeSup, DistribLattice.toLattice, instDistribLatticeOfLinearOrder, LinearOrder.toLattice, LinearOrder.toPartialOrder, Finset.card_singleton, lt_self_iff_false, Finset.sort_singleton, true_and, false_or] constructor · intro h unfold LE.le List.LE' Preorder.toLE PartialOrder.toPreorder LinearOrder.toPartialOrder List.instLinearOrder simp only [List.lex_lt, List.cons.injEq, and_true] if h' : a = b then exact Or.inl h' else right unfold LT.lt List.instLT simp only [List.lex_singleton_iff] exact lt_of_le_of_ne h h' · intro h unfold LE.le List.LE' Preorder.toLE PartialOrder.toPreorder LinearOrder.toPartialOrder List.instLinearOrder at h simp only [List.lex_lt, List.cons.injEq, and_true] at h rcases h with h | h · exact le_of_eq h · unfold LT.lt List.instLT at h simp only [List.lex_singleton_iff] at h exact le_of_lt h- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/OrderTheory/Lex'Order.lean:172-216
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Related declarations
Associated Primes ker mk Linear Map eq
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_ker_mkLinearMap_eq
Project documentation
Associated primes of the kernel of the localization map. This identifies the associated primes of ker (LocalizedModule.mkLinearMap S M) : Submodule R M with the associated primes of M that do meet the multiplicative set S. Equivalently, these are the associated primes of M after removing those disjoint from S. This lemma is used as the “kernel...
Source project: Harder-Narasimhan
Person-level attribution pending.
Associated Primes ker mk Linear Map subset
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_ker_mkLinearMap_subset
Plain-language statement
One-sided inclusion for associated primes of the kernel of the localization map. If p ∈ associatedPrimes R (ker (mkLinearMap S M)), then p is an associated prime of M and p is not disjoint from the multiplicative set S. This is one direction of associatedPrimes_ker_mkLinearMap_eq.
Source project: Harder-Narasimhan
Person-level attribution pending.
Associated Primes localized Module subset disjoint
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_localizedModule_subset_disjoint
Plain-language statement
Associated primes of a localized module are disjoint from the multiplicative set. More precisely, if p ∈ associatedPrimes R (LocalizedModule S M), then p.carrier ∩ S = ∅. This is a standard fact in commutative algebra: an element of S becomes a unit after localization, so no associated prime of the localized module can contain an element of S.
Source project: Harder-Narasimhan
Person-level attribution pending.